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Theorem prnzg 3492
 Description: A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
prnzg (𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)

Proof of Theorem prnzg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 preq1 3447 . . 3 (𝑥 = 𝐴 → {𝑥, 𝐵} = {𝐴, 𝐵})
21neeq1d 2223 . 2 (𝑥 = 𝐴 → ({𝑥, 𝐵} ≠ ∅ ↔ {𝐴, 𝐵} ≠ ∅))
3 vex 2560 . . 3 𝑥 ∈ V
43prnz 3490 . 2 {𝑥, 𝐵} ≠ ∅
52, 4vtoclg 2613 1 (𝐴𝑉 → {𝐴, 𝐵} ≠ ∅)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393   ≠ wne 2204  ∅c0 3224  {cpr 3376 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-un 2922  df-nul 3225  df-sn 3381  df-pr 3382 This theorem is referenced by:  0nelop  3985
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